(^152) The Three Theorems
and p be the velocity field and the density of a fluid in a region G. In our
discussion of divergence, we saw that div(pv) is the time rate of change of
the amount fluid in G per unit volume. Therefore, fff div(pv)dV is the
G
time rate of change of the amount of fluid in G, and must be equal to the
total flux across the boundary dG of G:
flJdiv(pv)dV= Ijpvdc
G dG
This is the essential meaning of the divergence theorem, which holds for
every continuously differentiable vector field F as stated below.
Theorem 3.2.2 Consider a bounded domain G in R^3 whose boundary
dG consists of finitely many piece-wise smooth orientable surfaces. Let F
be a vector field so that, for every unit vector u, the scalar field F • u has
a continuous gradient in an open set containing G. Then
ffF-dtr= fffdivFdV, (3.2.4)
dG G
where the normal vector to dG is pointing outside of G; if the boundary of
G consists of several pieces, then the integral on the left is the sum of the
corresponding integrals over all those pieces.
The great German mathematician CARL FRIEDRICH GAUSS (1777-
1855) discovered this result in 1813 while studying the laws of electrostatics.
As with many other his discoveries, he was not in a hurry to publish it; the
exact year is known from the detailed mathematical diary Gauss kept all his
adult life. Based on the entries in that diary, many believe that Gauss could
have advanced 19th century mathematics by another 50 years or more, had
he published all his results promptly. The Russian mathematician MIKHAIL
VASIL'EVICH OSTROGRADSKY (1801-1862) made an independent discovery
of (3.2.4) and published it around 1830.
EXERCISE 3.2.4? (a) Prove Gauss's Theorem. Hint: the argument is the same
as in the proof of Green's Theorem, but using (3.1.23). (b) Let F = r/r^3 = r/r^2
(see page 140). Let S be a piece-wise smooth closed surface so that the origin
O of the frame is not on the surface. Show that
II
F.d<r = C' S end°SeS thG °rigin' (3.2.5)
I 0, S does not enclose the origin.
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